![Page 1: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/1.jpg)
About Spetses
Michel Broue
Universite Paris-Diderot Paris VII
Buenos AiresJuly 2016
Michel Broue About Spetses
![Page 2: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/2.jpg)
An example of a known Unknown : Spets
Long lasting joint work between Gunter Malle, Jean Michel,and myself,
initiated on the Greek island named SPETSES in 1993
(there, we computed unipotent degrees, Frobenius eigenvalues, families, Fouriermatrix, for the “generic fake finite reductive group” (“Spets?”) whose Weylgroup is the cyclic group µ3 of order 3...),
going on with the collaboration of Olivier Dudas,and more and more of Cedric Bonnafe.
Michel Broue About Spetses
![Page 3: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/3.jpg)
An example of a known Unknown : Spets
Long lasting joint work between Gunter Malle, Jean Michel,and myself,
initiated on the Greek island named SPETSES in 1993
(there, we computed unipotent degrees, Frobenius eigenvalues, families, Fouriermatrix, for the “generic fake finite reductive group” (“Spets?”) whose Weylgroup is the cyclic group µ3 of order 3...),
going on with the collaboration of Olivier Dudas,and more and more of Cedric Bonnafe.
Michel Broue About Spetses
![Page 4: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/4.jpg)
An example of a known Unknown : Spets
Long lasting joint work between Gunter Malle, Jean Michel,and myself,
initiated on the Greek island named SPETSES in 1993
(there, we computed unipotent degrees, Frobenius eigenvalues, families, Fouriermatrix, for the “generic fake finite reductive group” (“Spets?”) whose Weylgroup is the cyclic group µ3 of order 3...),
going on with the collaboration of Olivier Dudas,and more and more of Cedric Bonnafe.
Michel Broue About Spetses
![Page 5: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/5.jpg)
An example of a known Unknown : Spets
Long lasting joint work between Gunter Malle, Jean Michel,and myself,
initiated on the Greek island named SPETSES in 1993
(there, we computed unipotent degrees, Frobenius eigenvalues, families, Fouriermatrix, for the “generic fake finite reductive group”
(“Spets?”) whose Weylgroup is the cyclic group µ3 of order 3...),
going on with the collaboration of Olivier Dudas,and more and more of Cedric Bonnafe.
Michel Broue About Spetses
![Page 6: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/6.jpg)
An example of a known Unknown : Spets
Long lasting joint work between Gunter Malle, Jean Michel,and myself,
initiated on the Greek island named SPETSES in 1993
(there, we computed unipotent degrees, Frobenius eigenvalues, families, Fouriermatrix, for the “generic fake finite reductive group” (“Spets?”)
whose Weylgroup is the cyclic group µ3 of order 3...),
going on with the collaboration of Olivier Dudas,and more and more of Cedric Bonnafe.
Michel Broue About Spetses
![Page 7: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/7.jpg)
An example of a known Unknown : Spets
Long lasting joint work between Gunter Malle, Jean Michel,and myself,
initiated on the Greek island named SPETSES in 1993
(there, we computed unipotent degrees, Frobenius eigenvalues, families, Fouriermatrix, for the “generic fake finite reductive group” (“Spets?”) whose Weylgroup is the cyclic group µ3 of order 3...),
going on with the collaboration of Olivier Dudas,and more and more of Cedric Bonnafe.
Michel Broue About Spetses
![Page 8: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/8.jpg)
An example of a known Unknown : Spets
Long lasting joint work between Gunter Malle, Jean Michel,and myself,
initiated on the Greek island named SPETSES in 1993
(there, we computed unipotent degrees, Frobenius eigenvalues, families, Fouriermatrix, for the “generic fake finite reductive group” (“Spets?”) whose Weylgroup is the cyclic group µ3 of order 3...),
going on with the collaboration of Olivier Dudas,and more and more of Cedric Bonnafe.
Michel Broue About Spetses
![Page 9: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/9.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 10: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/10.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 11: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/11.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),
with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 12: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/12.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W ,
group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 13: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/13.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y ,
endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 14: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/14.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 15: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/15.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 16: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/16.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))
is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 17: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/17.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 18: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/18.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture
we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 19: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/19.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 20: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/20.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 21: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/21.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 22: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/22.jpg)
I. Generic point of view on finite reductive groups
Let G be a connected reductive algebraic group over Fq
(e.g., G = GLn(Fp)),with Weyl group W , group of co-characters Y , endowed with a rationalstructure over Fq via a Frobenius endomorphism F .
The group G := GF
(e.g., G = GLn(q))is called a finite reductive group.
To simplify the lecture we assume that G is split, i.e., F acts on Y bymultiplication by q.
The type of G is G := (V , W ) where V := C⊗Z Y .
Lots of numerical data associated with G come from evaluation at x = qof polynomials in x which depends only on the type G.
Let us give some examples.
Michel Broue About Spetses
![Page 23: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/23.jpg)
1. The polynomial order
The element of Z[x ] defined by
|G|(x) = xN1
1
|W |∑
w∈W1
detV (1− wx)
is the polynomial order of G, that is, |G|(q) = |G | .Then
|G|(x) = xN∏
ζ mod Gal
Φζ(x)a(ζ) ,
Plus: for L an F -stable Levi subgroup of G, its porder |L|(x) divides |G|(x)).
I insist : the polynomial order depends only on the type (V ,W ) (noton a root datum). Thus
|SO2n+1(q)| = | Sp2n(q)| .
Michel Broue About Spetses
![Page 24: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/24.jpg)
1. The polynomial order
The element of Z[x ] defined by
|G|(x) = xN1
1
|W |∑
w∈W1
detV (1− wx)
is the polynomial order of G, that is, |G|(q) = |G | .Then
|G|(x) = xN∏
ζ mod Gal
Φζ(x)a(ζ) ,
Plus: for L an F -stable Levi subgroup of G, its porder |L|(x) divides |G|(x)).
I insist : the polynomial order depends only on the type (V ,W ) (noton a root datum). Thus
|SO2n+1(q)| = | Sp2n(q)| .
Michel Broue About Spetses
![Page 25: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/25.jpg)
1. The polynomial order
The element of Z[x ] defined by
|G|(x) = xN1
1
|W |∑
w∈W1
detV (1− wx)
is the polynomial order of G, that is, |G|(q) = |G | .Then
|G|(x) = xN∏
ζ mod Gal
Φζ(x)a(ζ) ,
Plus: for L an F -stable Levi subgroup of G, its porder |L|(x) divides |G|(x)).
I insist : the polynomial order depends only on the type (V ,W ) (noton a root datum). Thus
|SO2n+1(q)| = | Sp2n(q)| .
Michel Broue About Spetses
![Page 26: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/26.jpg)
1. The polynomial order
The element of Z[x ] defined by
|G|(x) = xN1
1
|W |∑
w∈W1
detV (1− wx)
is the polynomial order of G,
that is, |G|(q) = |G | .Then
|G|(x) = xN∏
ζ mod Gal
Φζ(x)a(ζ) ,
Plus: for L an F -stable Levi subgroup of G, its porder |L|(x) divides |G|(x)).
I insist : the polynomial order depends only on the type (V ,W ) (noton a root datum). Thus
|SO2n+1(q)| = | Sp2n(q)| .
Michel Broue About Spetses
![Page 27: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/27.jpg)
1. The polynomial order
The element of Z[x ] defined by
|G|(x) = xN1
1
|W |∑
w∈W1
detV (1− wx)
is the polynomial order of G, that is, |G|(q) = |G | .
Then|G|(x) = xN
∏ζ mod Gal
Φζ(x)a(ζ) ,
Plus: for L an F -stable Levi subgroup of G, its porder |L|(x) divides |G|(x)).
I insist : the polynomial order depends only on the type (V ,W ) (noton a root datum). Thus
|SO2n+1(q)| = | Sp2n(q)| .
Michel Broue About Spetses
![Page 28: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/28.jpg)
1. The polynomial order
The element of Z[x ] defined by
|G|(x) = xN1
1
|W |∑
w∈W1
detV (1− wx)
is the polynomial order of G, that is, |G|(q) = |G | .Then
|G|(x) = xN∏
ζ mod Gal
Φζ(x)a(ζ) ,
Plus: for L an F -stable Levi subgroup of G, its porder |L|(x) divides |G|(x)).
I insist : the polynomial order depends only on the type (V ,W ) (noton a root datum). Thus
|SO2n+1(q)| = | Sp2n(q)| .
Michel Broue About Spetses
![Page 29: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/29.jpg)
1. The polynomial order
The element of Z[x ] defined by
|G|(x) = xN1
1
|W |∑
w∈W1
detV (1− wx)
is the polynomial order of G, that is, |G|(q) = |G | .Then
|G|(x) = xN∏
ζ mod Gal
Φζ(x)a(ζ) ,
Plus: for L an F -stable Levi subgroup of G, its porder |L|(x) divides |G|(x)).
I insist : the polynomial order depends only on the type (V ,W ) (noton a root datum). Thus
|SO2n+1(q)| = | Sp2n(q)| .
Michel Broue About Spetses
![Page 30: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/30.jpg)
1. The polynomial order
The element of Z[x ] defined by
|G|(x) = xN1
1
|W |∑
w∈W1
detV (1− wx)
is the polynomial order of G, that is, |G|(q) = |G | .Then
|G|(x) = xN∏
ζ mod Gal
Φζ(x)a(ζ) ,
Plus: for L an F -stable Levi subgroup of G, its porder |L|(x) divides |G|(x)).
I insist : the polynomial order depends only on the type (V ,W ) (noton a root datum).
Thus
|SO2n+1(q)| = | Sp2n(q)| .
Michel Broue About Spetses
![Page 31: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/31.jpg)
1. The polynomial order
The element of Z[x ] defined by
|G|(x) = xN1
1
|W |∑
w∈W1
detV (1− wx)
is the polynomial order of G, that is, |G|(q) = |G | .Then
|G|(x) = xN∏
ζ mod Gal
Φζ(x)a(ζ) ,
Plus: for L an F -stable Levi subgroup of G, its porder |L|(x) divides |G|(x)).
I insist : the polynomial order depends only on the type (V ,W ) (noton a root datum). Thus
|SO2n+1(q)| = | Sp2n(q)| .
Michel Broue About Spetses
![Page 32: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/32.jpg)
2. Unipotent characters, generic degrees (Lusztig)
1 The set Un(G ) of unipotent characters of G is parametrized by theset of unipotent generic characters Un(G) (depending only on thetype). Let us denote that parametrization by
Un(G)→ Un(G ) , ρ 7→ ρq .
2 Generic degree : for all ρ ∈ Un(G), there exists Degρ(x) ∈ Q[x ] suchthat
Degρ(x) |x=q= Deg(ρq) = ρq(1) .
Of course, the (generic) degrees divide the (polynomial) order of G.
3 Every ρ ∈ Un(G) comes equipped with a Frobenius eigenvalue Frρ, aroot of unity...which has something to do with the Deligne–Lusztig varieties Xw ...
Michel Broue About Spetses
![Page 33: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/33.jpg)
2. Unipotent characters, generic degrees (Lusztig)
1 The set Un(G ) of unipotent characters of G is parametrized by theset of unipotent generic characters Un(G) (depending only on thetype). Let us denote that parametrization by
Un(G)→ Un(G ) , ρ 7→ ρq .
2 Generic degree : for all ρ ∈ Un(G), there exists Degρ(x) ∈ Q[x ] suchthat
Degρ(x) |x=q= Deg(ρq) = ρq(1) .
Of course, the (generic) degrees divide the (polynomial) order of G.
3 Every ρ ∈ Un(G) comes equipped with a Frobenius eigenvalue Frρ, aroot of unity...which has something to do with the Deligne–Lusztig varieties Xw ...
Michel Broue About Spetses
![Page 34: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/34.jpg)
2. Unipotent characters, generic degrees (Lusztig)
1 The set Un(G ) of unipotent characters of G is parametrized by theset of unipotent generic characters Un(G) (depending only on thetype). Let us denote that parametrization by
Un(G)→ Un(G ) , ρ 7→ ρq .
2 Generic degree : for all ρ ∈ Un(G), there exists Degρ(x) ∈ Q[x ] suchthat
Degρ(x) |x=q= Deg(ρq)
= ρq(1) .
Of course, the (generic) degrees divide the (polynomial) order of G.
3 Every ρ ∈ Un(G) comes equipped with a Frobenius eigenvalue Frρ, aroot of unity...which has something to do with the Deligne–Lusztig varieties Xw ...
Michel Broue About Spetses
![Page 35: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/35.jpg)
2. Unipotent characters, generic degrees (Lusztig)
1 The set Un(G ) of unipotent characters of G is parametrized by theset of unipotent generic characters Un(G) (depending only on thetype). Let us denote that parametrization by
Un(G)→ Un(G ) , ρ 7→ ρq .
2 Generic degree : for all ρ ∈ Un(G), there exists Degρ(x) ∈ Q[x ] suchthat
Degρ(x) |x=q= Deg(ρq) = ρq(1) .
Of course, the (generic) degrees divide the (polynomial) order of G.
3 Every ρ ∈ Un(G) comes equipped with a Frobenius eigenvalue Frρ, aroot of unity...which has something to do with the Deligne–Lusztig varieties Xw ...
Michel Broue About Spetses
![Page 36: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/36.jpg)
2. Unipotent characters, generic degrees (Lusztig)
1 The set Un(G ) of unipotent characters of G is parametrized by theset of unipotent generic characters Un(G) (depending only on thetype). Let us denote that parametrization by
Un(G)→ Un(G ) , ρ 7→ ρq .
2 Generic degree : for all ρ ∈ Un(G), there exists Degρ(x) ∈ Q[x ] suchthat
Degρ(x) |x=q= Deg(ρq) = ρq(1) .
Of course, the (generic) degrees divide the (polynomial) order of G.
3 Every ρ ∈ Un(G) comes equipped with a Frobenius eigenvalue Frρ, aroot of unity...which has something to do with the Deligne–Lusztig varieties Xw ...
Michel Broue About Spetses
![Page 37: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/37.jpg)
2. Unipotent characters, generic degrees (Lusztig)
1 The set Un(G ) of unipotent characters of G is parametrized by theset of unipotent generic characters Un(G) (depending only on thetype). Let us denote that parametrization by
Un(G)→ Un(G ) , ρ 7→ ρq .
2 Generic degree : for all ρ ∈ Un(G), there exists Degρ(x) ∈ Q[x ] suchthat
Degρ(x) |x=q= Deg(ρq) = ρq(1) .
Of course, the (generic) degrees divide the (polynomial) order of G.
3 Every ρ ∈ Un(G) comes equipped with a Frobenius eigenvalue Frρ, aroot of unity
...which has something to do with the Deligne–Lusztig varieties Xw ...
Michel Broue About Spetses
![Page 38: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/38.jpg)
2. Unipotent characters, generic degrees (Lusztig)
1 The set Un(G ) of unipotent characters of G is parametrized by theset of unipotent generic characters Un(G) (depending only on thetype). Let us denote that parametrization by
Un(G)→ Un(G ) , ρ 7→ ρq .
2 Generic degree : for all ρ ∈ Un(G), there exists Degρ(x) ∈ Q[x ] suchthat
Degρ(x) |x=q= Deg(ρq) = ρq(1) .
Of course, the (generic) degrees divide the (polynomial) order of G.
3 Every ρ ∈ Un(G) comes equipped with a Frobenius eigenvalue Frρ, aroot of unity...which has something to do with the Deligne–Lusztig varieties Xw ...
Michel Broue About Spetses
![Page 39: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/39.jpg)
Generic unipotent characters, continued
... have lots of other properties, like the partition of Un(G) intoHarish-Chandra series, or more generally
5 For all ζ ∈ µ, partition of Un(G) into ζ-Harish-Chandra series.
6 Description of the principal ζ-Harish-Chandra series with aζ-cyclotomic Hecke algebra.
The principal 1-Harish-Chandra series is the usual principal Harish-Chandra series.
Let us devote some time to the notion of ζ-cyclotomic Heckealgebras.Then we shall come back to the generic properties of Un(G).
Michel Broue About Spetses
![Page 40: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/40.jpg)
Generic unipotent characters, continued
... have lots of other properties, like the partition of Un(G) intoHarish-Chandra series, or more generally
5 For all ζ ∈ µ, partition of Un(G) into ζ-Harish-Chandra series.
6 Description of the principal ζ-Harish-Chandra series with aζ-cyclotomic Hecke algebra.
The principal 1-Harish-Chandra series is the usual principal Harish-Chandra series.
Let us devote some time to the notion of ζ-cyclotomic Heckealgebras.Then we shall come back to the generic properties of Un(G).
Michel Broue About Spetses
![Page 41: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/41.jpg)
Generic unipotent characters, continued
... have lots of other properties, like the partition of Un(G) intoHarish-Chandra series, or more generally
5 For all ζ ∈ µ, partition of Un(G) into ζ-Harish-Chandra series.
6 Description of the principal ζ-Harish-Chandra series with aζ-cyclotomic Hecke algebra.
The principal 1-Harish-Chandra series is the usual principal Harish-Chandra series.
Let us devote some time to the notion of ζ-cyclotomic Heckealgebras.Then we shall come back to the generic properties of Un(G).
Michel Broue About Spetses
![Page 42: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/42.jpg)
Generic unipotent characters, continued
... have lots of other properties, like the partition of Un(G) intoHarish-Chandra series, or more generally
5 For all ζ ∈ µ, partition of Un(G) into ζ-Harish-Chandra series.
6 Description of the principal ζ-Harish-Chandra series with aζ-cyclotomic Hecke algebra.
The principal 1-Harish-Chandra series is the usual principal Harish-Chandra series.
Let us devote some time to the notion of ζ-cyclotomic Heckealgebras.Then we shall come back to the generic properties of Un(G).
Michel Broue About Spetses
![Page 43: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/43.jpg)
Generic unipotent characters, continued
... have lots of other properties, like the partition of Un(G) intoHarish-Chandra series, or more generally
5 For all ζ ∈ µ, partition of Un(G) into ζ-Harish-Chandra series.
6 Description of the principal ζ-Harish-Chandra series with aζ-cyclotomic Hecke algebra.
The principal 1-Harish-Chandra series is the usual principal Harish-Chandra series.
Let us devote some time to the notion of ζ-cyclotomic Heckealgebras.Then we shall come back to the generic properties of Un(G).
Michel Broue About Spetses
![Page 44: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/44.jpg)
Generic unipotent characters, continued
... have lots of other properties, like the partition of Un(G) intoHarish-Chandra series, or more generally
5 For all ζ ∈ µ, partition of Un(G) into ζ-Harish-Chandra series.
6 Description of the principal ζ-Harish-Chandra series with aζ-cyclotomic Hecke algebra.
The principal 1-Harish-Chandra series is the usual principal Harish-Chandra series.
Let us devote some time to the notion of ζ-cyclotomic Heckealgebras.
Then we shall come back to the generic properties of Un(G).
Michel Broue About Spetses
![Page 45: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/45.jpg)
Generic unipotent characters, continued
... have lots of other properties, like the partition of Un(G) intoHarish-Chandra series, or more generally
5 For all ζ ∈ µ, partition of Un(G) into ζ-Harish-Chandra series.
6 Description of the principal ζ-Harish-Chandra series with aζ-cyclotomic Hecke algebra.
The principal 1-Harish-Chandra series is the usual principal Harish-Chandra series.
Let us devote some time to the notion of ζ-cyclotomic Heckealgebras.Then we shall come back to the generic properties of Un(G).
Michel Broue About Spetses
![Page 46: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/46.jpg)
3. ζ-regular elements and ζ-cyclotomic Hecke algebras
What follows holds more generally for any pair G = (V ,W ) where
V is a finite dimensional complex vector space,
W is a finite subgroup of GL(V ) generated by (pseudo)-reflections.
Let A be the set of reflecting hyperplanes of W . A root of unity ζ iscalled regular if there exist w ∈W and x ∈ V reg := V \
⋃H∈AH such
that w(x) = ζx . We then say that w is ζ-regular.
From now on we assume that ζ is regular. Then
[Springer] The group Wζ := CW (w) acts faithfully as a reflectiongroup on the vector space Vζ := ker(w − ζIdV ).
The group Wζ is called the ζ-cyclotomic Weyl group.
[Note that W1 = W ].
Michel Broue About Spetses
![Page 47: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/47.jpg)
3. ζ-regular elements and ζ-cyclotomic Hecke algebras
What follows holds more generally for any pair G = (V ,W ) where
V is a finite dimensional complex vector space,
W is a finite subgroup of GL(V ) generated by (pseudo)-reflections.
Let A be the set of reflecting hyperplanes of W . A root of unity ζ iscalled regular if there exist w ∈W and x ∈ V reg := V \
⋃H∈AH such
that w(x) = ζx . We then say that w is ζ-regular.
From now on we assume that ζ is regular. Then
[Springer] The group Wζ := CW (w) acts faithfully as a reflectiongroup on the vector space Vζ := ker(w − ζIdV ).
The group Wζ is called the ζ-cyclotomic Weyl group.
[Note that W1 = W ].
Michel Broue About Spetses
![Page 48: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/48.jpg)
3. ζ-regular elements and ζ-cyclotomic Hecke algebras
What follows holds more generally for any pair G = (V ,W ) where
V is a finite dimensional complex vector space,
W is a finite subgroup of GL(V ) generated by (pseudo)-reflections.
Let A be the set of reflecting hyperplanes of W . A root of unity ζ iscalled regular if there exist w ∈W and x ∈ V reg := V \
⋃H∈AH such
that w(x) = ζx . We then say that w is ζ-regular.
From now on we assume that ζ is regular. Then
[Springer] The group Wζ := CW (w) acts faithfully as a reflectiongroup on the vector space Vζ := ker(w − ζIdV ).
The group Wζ is called the ζ-cyclotomic Weyl group.
[Note that W1 = W ].
Michel Broue About Spetses
![Page 49: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/49.jpg)
3. ζ-regular elements and ζ-cyclotomic Hecke algebras
What follows holds more generally for any pair G = (V ,W ) where
V is a finite dimensional complex vector space,
W is a finite subgroup of GL(V ) generated by (pseudo)-reflections.
Let A be the set of reflecting hyperplanes of W . A root of unity ζ iscalled regular if there exist w ∈W and x ∈ V reg := V \
⋃H∈AH such
that w(x) = ζx . We then say that w is ζ-regular.
From now on we assume that ζ is regular. Then
[Springer] The group Wζ := CW (w) acts faithfully as a reflectiongroup on the vector space Vζ := ker(w − ζIdV ).
The group Wζ is called the ζ-cyclotomic Weyl group.
[Note that W1 = W ].
Michel Broue About Spetses
![Page 50: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/50.jpg)
3. ζ-regular elements and ζ-cyclotomic Hecke algebras
What follows holds more generally for any pair G = (V ,W ) where
V is a finite dimensional complex vector space,
W is a finite subgroup of GL(V ) generated by (pseudo)-reflections.
Let A be the set of reflecting hyperplanes of W . A root of unity ζ iscalled regular if there exist w ∈W and x ∈ V reg := V \
⋃H∈AH such
that w(x) = ζx . We then say that w is ζ-regular.
From now on we assume that ζ is regular. Then
[Springer] The group Wζ := CW (w) acts faithfully as a reflectiongroup on the vector space Vζ := ker(w − ζIdV ).
The group Wζ is called the ζ-cyclotomic Weyl group.
[Note that W1 = W ].
Michel Broue About Spetses
![Page 51: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/51.jpg)
3. ζ-regular elements and ζ-cyclotomic Hecke algebras
What follows holds more generally for any pair G = (V ,W ) where
V is a finite dimensional complex vector space,
W is a finite subgroup of GL(V ) generated by (pseudo)-reflections.
Let A be the set of reflecting hyperplanes of W . A root of unity ζ iscalled regular if there exist w ∈W and x ∈ V reg := V \
⋃H∈AH such
that w(x) = ζx . We then say that w is ζ-regular.
From now on we assume that ζ is regular. Then
[Springer] The group Wζ := CW (w) acts faithfully as a reflectiongroup on the vector space Vζ := ker(w − ζIdV ).
The group Wζ is called the ζ-cyclotomic Weyl group.
[Note that W1 = W ].
Michel Broue About Spetses
![Page 52: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/52.jpg)
3. ζ-regular elements and ζ-cyclotomic Hecke algebras
What follows holds more generally for any pair G = (V ,W ) where
V is a finite dimensional complex vector space,
W is a finite subgroup of GL(V ) generated by (pseudo)-reflections.
Let A be the set of reflecting hyperplanes of W . A root of unity ζ iscalled regular if there exist w ∈W and x ∈ V reg := V \
⋃H∈AH such
that w(x) = ζx . We then say that w is ζ-regular.
From now on we assume that ζ is regular. Then
[Springer] The group Wζ := CW (w) acts faithfully as a reflectiongroup on the vector space Vζ := ker(w − ζIdV ).
The group Wζ is called the ζ-cyclotomic Weyl group.
[Note that W1 = W ].
Michel Broue About Spetses
![Page 53: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/53.jpg)
3. ζ-regular elements and ζ-cyclotomic Hecke algebras
What follows holds more generally for any pair G = (V ,W ) where
V is a finite dimensional complex vector space,
W is a finite subgroup of GL(V ) generated by (pseudo)-reflections.
Let A be the set of reflecting hyperplanes of W . A root of unity ζ iscalled regular if there exist w ∈W and x ∈ V reg := V \
⋃H∈AH such
that w(x) = ζx . We then say that w is ζ-regular.
From now on we assume that ζ is regular. Then
[Springer] The group Wζ := CW (w) acts faithfully as a reflectiongroup on the vector space Vζ := ker(w − ζIdV ).
The group Wζ is called the ζ-cyclotomic Weyl group.
[Note that W1 = W ].
Michel Broue About Spetses
![Page 54: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/54.jpg)
3. ζ-regular elements and ζ-cyclotomic Hecke algebras
What follows holds more generally for any pair G = (V ,W ) where
V is a finite dimensional complex vector space,
W is a finite subgroup of GL(V ) generated by (pseudo)-reflections.
Let A be the set of reflecting hyperplanes of W . A root of unity ζ iscalled regular if there exist w ∈W and x ∈ V reg := V \
⋃H∈AH such
that w(x) = ζx . We then say that w is ζ-regular.
From now on we assume that ζ is regular. Then
[Springer] The group Wζ := CW (w) acts faithfully as a reflectiongroup on the vector space Vζ := ker(w − ζIdV ).
The group Wζ is called the ζ-cyclotomic Weyl group.
[Note that W1 = W ].
Michel Broue About Spetses
![Page 55: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/55.jpg)
3.1. Hecke algebras everywhere
One knows that the (ordinary) principal series Un(G, 1)
corresponds to unipotent characters of G occurring in Q`(G/B),
and the commutant of that module is the (ordinary) Hecke algebra ofW evaluated at q.
One conjectures that the ζ–principal series Un(G, ζ)n! for a good choice of w
corresponds to unipotent characters of G occurring in⊕n H
nc (Xw ,Q`),
and the commutant of that module is a ζ-cyclotomic Hecke algebraof Wζ evaluated at q.
We shall now introduce the notion of ζ-cyclotomic Hecke algebra.
Michel Broue About Spetses
![Page 56: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/56.jpg)
3.1. Hecke algebras everywhere
One knows that the (ordinary) principal series Un(G, 1)
corresponds to unipotent characters of G occurring in Q`(G/B),
and the commutant of that module is the (ordinary) Hecke algebra ofW evaluated at q.
One conjectures that the ζ–principal series Un(G, ζ)n! for a good choice of w
corresponds to unipotent characters of G occurring in⊕n H
nc (Xw ,Q`),
and the commutant of that module is a ζ-cyclotomic Hecke algebraof Wζ evaluated at q.
We shall now introduce the notion of ζ-cyclotomic Hecke algebra.
Michel Broue About Spetses
![Page 57: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/57.jpg)
3.1. Hecke algebras everywhere
One knows that the (ordinary) principal series Un(G, 1)
corresponds to unipotent characters of G occurring in Q`(G/B),
and the commutant of that module is the (ordinary) Hecke algebra ofW evaluated at q.
One conjectures that the ζ–principal series Un(G, ζ)n! for a good choice of w
corresponds to unipotent characters of G occurring in⊕n H
nc (Xw ,Q`),
and the commutant of that module is a ζ-cyclotomic Hecke algebraof Wζ evaluated at q.
We shall now introduce the notion of ζ-cyclotomic Hecke algebra.
Michel Broue About Spetses
![Page 58: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/58.jpg)
3.1. Hecke algebras everywhere
One knows that the (ordinary) principal series Un(G, 1)
corresponds to unipotent characters of G occurring in Q`(G/B),
and the commutant of that module is the (ordinary) Hecke algebra ofW evaluated at q.
One conjectures that the ζ–principal series Un(G, ζ)n! for a good choice of w
corresponds to unipotent characters of G occurring in⊕n H
nc (Xw ,Q`),
and the commutant of that module is a ζ-cyclotomic Hecke algebraof Wζ evaluated at q.
We shall now introduce the notion of ζ-cyclotomic Hecke algebra.
Michel Broue About Spetses
![Page 59: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/59.jpg)
3.1. Hecke algebras everywhere
One knows that the (ordinary) principal series Un(G, 1)
corresponds to unipotent characters of G occurring in Q`(G/B),
and the commutant of that module is the (ordinary) Hecke algebra ofW evaluated at q.
One conjectures that the ζ–principal series Un(G, ζ)
n! for a good choice of w
corresponds to unipotent characters of G occurring in⊕n H
nc (Xw ,Q`),
and the commutant of that module is a ζ-cyclotomic Hecke algebraof Wζ evaluated at q.
We shall now introduce the notion of ζ-cyclotomic Hecke algebra.
Michel Broue About Spetses
![Page 60: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/60.jpg)
3.1. Hecke algebras everywhere
One knows that the (ordinary) principal series Un(G, 1)
corresponds to unipotent characters of G occurring in Q`(G/B),
and the commutant of that module is the (ordinary) Hecke algebra ofW evaluated at q.
One conjectures that the ζ–principal series Un(G, ζ)n! for a good choice of w
corresponds to unipotent characters of G occurring in⊕n H
nc (Xw ,Q`),
and the commutant of that module is a ζ-cyclotomic Hecke algebraof Wζ evaluated at q.
We shall now introduce the notion of ζ-cyclotomic Hecke algebra.
Michel Broue About Spetses
![Page 61: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/61.jpg)
3.1. Hecke algebras everywhere
One knows that the (ordinary) principal series Un(G, 1)
corresponds to unipotent characters of G occurring in Q`(G/B),
and the commutant of that module is the (ordinary) Hecke algebra ofW evaluated at q.
One conjectures that the ζ–principal series Un(G, ζ)n! for a good choice of w
corresponds to unipotent characters of G occurring in⊕n H
nc (Xw ,Q`),
and the commutant of that module is a ζ-cyclotomic Hecke algebraof Wζ evaluated at q.
We shall now introduce the notion of ζ-cyclotomic Hecke algebra.
Michel Broue About Spetses
![Page 62: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/62.jpg)
3.1. Hecke algebras everywhere
One knows that the (ordinary) principal series Un(G, 1)
corresponds to unipotent characters of G occurring in Q`(G/B),
and the commutant of that module is the (ordinary) Hecke algebra ofW evaluated at q.
One conjectures that the ζ–principal series Un(G, ζ)n! for a good choice of w
corresponds to unipotent characters of G occurring in⊕n H
nc (Xw ,Q`),
and the commutant of that module is a ζ-cyclotomic Hecke algebraof Wζ evaluated at q.
We shall now introduce the notion of ζ-cyclotomic Hecke algebra.
Michel Broue About Spetses
![Page 63: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/63.jpg)
3.1. Hecke algebras everywhere
One knows that the (ordinary) principal series Un(G, 1)
corresponds to unipotent characters of G occurring in Q`(G/B),
and the commutant of that module is the (ordinary) Hecke algebra ofW evaluated at q.
One conjectures that the ζ–principal series Un(G, ζ)n! for a good choice of w
corresponds to unipotent characters of G occurring in⊕n H
nc (Xw ,Q`),
and the commutant of that module is a ζ-cyclotomic Hecke algebraof Wζ evaluated at q.
We shall now introduce the notion of ζ-cyclotomic Hecke algebra.
Michel Broue About Spetses
![Page 64: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/64.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©
s©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 65: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/65.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particular
I a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©
s©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 66: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/66.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,
I an image of the group algebra of the braid group BWζattached to Wζ ,
I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©
s©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 67: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/67.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,
I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©
s©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 68: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/68.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,I a deformation (via x) of the group algebra of Wζ ,
I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©
s©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 69: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/69.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©
s©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 70: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/70.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:
I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©s
©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 71: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/71.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3
←→ ©s
©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 72: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/72.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©
s©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 73: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/73.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©
s©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 74: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/74.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©
s©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2)
←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 75: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/75.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©
s©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 76: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/76.jpg)
3.2. ζ-cyclotomic Hecke algebras
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is in particularI a C[x , x−1]-algebra,I an image of the group algebra of the braid group BWζ
attached to Wζ ,I a deformation (via x) of the group algebra of Wζ ,I which specializes to that algebra for x = ζ.
Examples:I Case where G = GL3, ζ = 1 : Wζ = W = S3 ←→ ©
s©t
H(W ) =
⟨S ,T ; STS = TST , (S − x)(S + 1) = 0
⟩is 1-cyclotomic.
I For G = O8(q), W = D4, ζ = i , Wi = G(4, 2, 2) ←→ s©2 n©2 t
©2 u
H(Wi ) =
⟨S ,T ,U ;
STU = TUS = UST
(S − x2)(S − 1) = 0
⟩
Michel Broue About Spetses
![Page 77: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/77.jpg)
Fundamental properties Case by case checking...
“There is a proof, but so far I’ve not seen an explanation” [JHC]
Such an algebra has a canonical symmetrizing form τ .
It becomes split semisimple over C(x1/|ZWζ |).
⇒ Hence each absolute irreducible character χ of H(Wζ) is equippedwith a Schur element
Sχ ∈ C[x1/|ZWζ |, x−1/|ZWζ |] defined by τ =∑
χ∈IrrH(Wζ)
χ
Sχ.
Michel Broue About Spetses
![Page 78: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/78.jpg)
Fundamental properties Case by case checking...
“There is a proof, but so far I’ve not seen an explanation” [JHC]
Such an algebra has a canonical symmetrizing form τ .
It becomes split semisimple over C(x1/|ZWζ |).
⇒ Hence each absolute irreducible character χ of H(Wζ) is equippedwith a Schur element
Sχ ∈ C[x1/|ZWζ |, x−1/|ZWζ |] defined by τ =∑
χ∈IrrH(Wζ)
χ
Sχ.
Michel Broue About Spetses
![Page 79: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/79.jpg)
Fundamental properties Case by case checking...
“There is a proof, but so far I’ve not seen an explanation” [JHC]
Such an algebra has a canonical symmetrizing form τ .
It becomes split semisimple over C(x1/|ZWζ |).
⇒ Hence each absolute irreducible character χ of H(Wζ) is equippedwith a Schur element
Sχ ∈ C[x1/|ZWζ |, x−1/|ZWζ |] defined by τ =∑
χ∈IrrH(Wζ)
χ
Sχ.
Michel Broue About Spetses
![Page 80: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/80.jpg)
Fundamental properties Case by case checking...
“There is a proof, but so far I’ve not seen an explanation” [JHC]
Such an algebra has a canonical symmetrizing form τ .
It becomes split semisimple over C(x1/|ZWζ |).
⇒ Hence each absolute irreducible character χ of H(Wζ) is equippedwith a Schur element
Sχ ∈ C[x1/|ZWζ |, x−1/|ZWζ |] defined by τ =∑
χ∈IrrH(Wζ)
χ
Sχ.
Michel Broue About Spetses
![Page 81: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/81.jpg)
Fundamental properties Case by case checking...
“There is a proof, but so far I’ve not seen an explanation” [JHC]
Such an algebra has a canonical symmetrizing form τ .
It becomes split semisimple over C(x1/|ZWζ |).
⇒ Hence each absolute irreducible character χ of H(Wζ) is equippedwith a Schur element
Sχ ∈ C[x1/|ZWζ |, x−1/|ZWζ |] defined by τ =∑
χ∈IrrH(Wζ)
χ
Sχ.
Michel Broue About Spetses
![Page 82: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/82.jpg)
Fundamental properties Case by case checking...
“There is a proof, but so far I’ve not seen an explanation” [JHC]
Such an algebra has a canonical symmetrizing form τ .
It becomes split semisimple over C(x1/|ZWζ |).
⇒ Hence each absolute irreducible character χ of H(Wζ) is equippedwith a Schur element
Sχ ∈ C[x1/|ZWζ |, x−1/|ZWζ |] defined by τ =∑
χ∈IrrH(Wζ)
χ
Sχ.
Michel Broue About Spetses
![Page 83: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/83.jpg)
3.3. Spetsial ζ-cyclotomic Hecke algebras
Definition
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is spetsial for G if
1 it satisfies various technical conditions...
2 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1]
(and not only C[x1/|ZWζ |, x−1/|ZWζ |]).
Michel Broue About Spetses
![Page 84: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/84.jpg)
3.3. Spetsial ζ-cyclotomic Hecke algebras
Definition
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is spetsial for G if
1 it satisfies various technical conditions...
2 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1]
(and not only C[x1/|ZWζ |, x−1/|ZWζ |]).
Michel Broue About Spetses
![Page 85: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/85.jpg)
3.3. Spetsial ζ-cyclotomic Hecke algebras
Definition
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is spetsial for G if
1 it satisfies various technical conditions...
2 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1]
(and not only C[x1/|ZWζ |, x−1/|ZWζ |]).
Michel Broue About Spetses
![Page 86: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/86.jpg)
3.3. Spetsial ζ-cyclotomic Hecke algebras
Definition
A ζ-cyclotomic Hecke algebra H(Wζ) of Wζ is spetsial for G if
1 it satisfies various technical conditions...
2 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1]
(and not only C[x1/|ZWζ |, x−1/|ZWζ |]).
Michel Broue About Spetses
![Page 87: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/87.jpg)
On Un(G) again
When we started speaking about ζ-regular elements, ζ-cyclotomic Weyl groups, spetsialζ-cyclotomic Hecke algebras, we were stating “generic properties” of unipotentcharacters :... lots of other properties, like the partition of Un(G) into ζ-Harish-Chandra series:
6 Description of the principal ζ-Harish-Chandra series with a spetsial ζ-cyclotomicHecke algebra.
Let us come back to that long list.
7 Partition of Un(G) into families and their Intersections withζ-Harish-Chandra series (Rouquier blocks).
8 Ennola permutation on Un(G).[This is an abstract formulation of the fact that Un(q) = ±GLn(−q)]
9 Fourier matrices and SL2(Z)-representation.
We shall review this now in the more general context of “Spetses”.
Michel Broue About Spetses
![Page 88: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/88.jpg)
On Un(G) again
When we started speaking about ζ-regular elements, ζ-cyclotomic Weyl groups, spetsialζ-cyclotomic Hecke algebras, we were stating “generic properties” of unipotentcharacters :
... lots of other properties, like the partition of Un(G) into ζ-Harish-Chandra series:
6 Description of the principal ζ-Harish-Chandra series with a spetsial ζ-cyclotomicHecke algebra.
Let us come back to that long list.
7 Partition of Un(G) into families and their Intersections withζ-Harish-Chandra series (Rouquier blocks).
8 Ennola permutation on Un(G).[This is an abstract formulation of the fact that Un(q) = ±GLn(−q)]
9 Fourier matrices and SL2(Z)-representation.
We shall review this now in the more general context of “Spetses”.
Michel Broue About Spetses
![Page 89: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/89.jpg)
On Un(G) again
When we started speaking about ζ-regular elements, ζ-cyclotomic Weyl groups, spetsialζ-cyclotomic Hecke algebras, we were stating “generic properties” of unipotentcharacters :... lots of other properties, like the partition of Un(G) into ζ-Harish-Chandra series:
6 Description of the principal ζ-Harish-Chandra series with a spetsial ζ-cyclotomicHecke algebra.
Let us come back to that long list.
7 Partition of Un(G) into families and their Intersections withζ-Harish-Chandra series (Rouquier blocks).
8 Ennola permutation on Un(G).[This is an abstract formulation of the fact that Un(q) = ±GLn(−q)]
9 Fourier matrices and SL2(Z)-representation.
We shall review this now in the more general context of “Spetses”.
Michel Broue About Spetses
![Page 90: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/90.jpg)
On Un(G) again
When we started speaking about ζ-regular elements, ζ-cyclotomic Weyl groups, spetsialζ-cyclotomic Hecke algebras, we were stating “generic properties” of unipotentcharacters :... lots of other properties, like the partition of Un(G) into ζ-Harish-Chandra series:
6 Description of the principal ζ-Harish-Chandra series with a spetsial ζ-cyclotomicHecke algebra.
Let us come back to that long list.
7 Partition of Un(G) into families and their Intersections withζ-Harish-Chandra series (Rouquier blocks).
8 Ennola permutation on Un(G).[This is an abstract formulation of the fact that Un(q) = ±GLn(−q)]
9 Fourier matrices and SL2(Z)-representation.
We shall review this now in the more general context of “Spetses”.
Michel Broue About Spetses
![Page 91: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/91.jpg)
On Un(G) again
When we started speaking about ζ-regular elements, ζ-cyclotomic Weyl groups, spetsialζ-cyclotomic Hecke algebras, we were stating “generic properties” of unipotentcharacters :... lots of other properties, like the partition of Un(G) into ζ-Harish-Chandra series:
6 Description of the principal ζ-Harish-Chandra series with a spetsial ζ-cyclotomicHecke algebra.
Let us come back to that long list.
7 Partition of Un(G) into families and their Intersections withζ-Harish-Chandra series (Rouquier blocks).
8 Ennola permutation on Un(G).[This is an abstract formulation of the fact that Un(q) = ±GLn(−q)]
9 Fourier matrices and SL2(Z)-representation.
We shall review this now in the more general context of “Spetses”.
Michel Broue About Spetses
![Page 92: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/92.jpg)
On Un(G) again
When we started speaking about ζ-regular elements, ζ-cyclotomic Weyl groups, spetsialζ-cyclotomic Hecke algebras, we were stating “generic properties” of unipotentcharacters :... lots of other properties, like the partition of Un(G) into ζ-Harish-Chandra series:
6 Description of the principal ζ-Harish-Chandra series with a spetsial ζ-cyclotomicHecke algebra.
Let us come back to that long list.
7 Partition of Un(G) into families and their Intersections withζ-Harish-Chandra series (Rouquier blocks).
8 Ennola permutation on Un(G).
[This is an abstract formulation of the fact that Un(q) = ±GLn(−q)]
9 Fourier matrices and SL2(Z)-representation.
We shall review this now in the more general context of “Spetses”.
Michel Broue About Spetses
![Page 93: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/93.jpg)
On Un(G) again
When we started speaking about ζ-regular elements, ζ-cyclotomic Weyl groups, spetsialζ-cyclotomic Hecke algebras, we were stating “generic properties” of unipotentcharacters :... lots of other properties, like the partition of Un(G) into ζ-Harish-Chandra series:
6 Description of the principal ζ-Harish-Chandra series with a spetsial ζ-cyclotomicHecke algebra.
Let us come back to that long list.
7 Partition of Un(G) into families and their Intersections withζ-Harish-Chandra series (Rouquier blocks).
8 Ennola permutation on Un(G).[This is an abstract formulation of the fact that Un(q) = ±GLn(−q)]
9 Fourier matrices and SL2(Z)-representation.
We shall review this now in the more general context of “Spetses”.
Michel Broue About Spetses
![Page 94: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/94.jpg)
On Un(G) again
When we started speaking about ζ-regular elements, ζ-cyclotomic Weyl groups, spetsialζ-cyclotomic Hecke algebras, we were stating “generic properties” of unipotentcharacters :... lots of other properties, like the partition of Un(G) into ζ-Harish-Chandra series:
6 Description of the principal ζ-Harish-Chandra series with a spetsial ζ-cyclotomicHecke algebra.
Let us come back to that long list.
7 Partition of Un(G) into families and their Intersections withζ-Harish-Chandra series (Rouquier blocks).
8 Ennola permutation on Un(G).[This is an abstract formulation of the fact that Un(q) = ±GLn(−q)]
9 Fourier matrices and SL2(Z)-representation.
We shall review this now in the more general context of “Spetses”.
Michel Broue About Spetses
![Page 95: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/95.jpg)
On Un(G) again
When we started speaking about ζ-regular elements, ζ-cyclotomic Weyl groups, spetsialζ-cyclotomic Hecke algebras, we were stating “generic properties” of unipotentcharacters :... lots of other properties, like the partition of Un(G) into ζ-Harish-Chandra series:
6 Description of the principal ζ-Harish-Chandra series with a spetsial ζ-cyclotomicHecke algebra.
Let us come back to that long list.
7 Partition of Un(G) into families and their Intersections withζ-Harish-Chandra series (Rouquier blocks).
8 Ennola permutation on Un(G).[This is an abstract formulation of the fact that Un(q) = ±GLn(−q)]
9 Fourier matrices and SL2(Z)-representation.
We shall review this now in the more general context of “Spetses”.
Michel Broue About Spetses
![Page 96: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/96.jpg)
II. Towards Spetses
Try to treat a complex reflection group as a Weyl group: try to builda thing G(x) (x an indeterminate) associated with a typeG = (V ,W ) where W is a (pseudo)-reflection group.
Try at least to build “unipotent characters” of G, or at least to buildtheir degrees (polynomials in x), Frobenius eigenvalues (roots ofunity), Fourier matrices.
I Lusztig knew already a solution for Coxeter groups which are not Weylgroups (except the Fourier matrix for H4 which was determined byMalle in 1994).
I Malle gave a solution for imprimitive spetsial complex reflectiongroups in 1995.
I Stating a long series of precise axioms — many of technical nature —we can now show that there is a unique solution for all primitive
spetsial complex reflection groups.
M. Broue, G. Malle, J. Michel, Split spetses for primitive reflection groups, Asterisque 359 (2014)
Michel Broue About Spetses
![Page 97: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/97.jpg)
II. Towards Spetses
Try to treat a complex reflection group as a Weyl group: try to builda thing G(x) (x an indeterminate) associated with a typeG = (V ,W ) where W is a (pseudo)-reflection group.
Try at least to build “unipotent characters” of G, or at least to buildtheir degrees (polynomials in x), Frobenius eigenvalues (roots ofunity), Fourier matrices.
I Lusztig knew already a solution for Coxeter groups which are not Weylgroups (except the Fourier matrix for H4 which was determined byMalle in 1994).
I Malle gave a solution for imprimitive spetsial complex reflectiongroups in 1995.
I Stating a long series of precise axioms — many of technical nature —we can now show that there is a unique solution for all primitive
spetsial complex reflection groups.
M. Broue, G. Malle, J. Michel, Split spetses for primitive reflection groups, Asterisque 359 (2014)
Michel Broue About Spetses
![Page 98: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/98.jpg)
II. Towards Spetses
Try to treat a complex reflection group as a Weyl group: try to builda thing G(x) (x an indeterminate) associated with a typeG = (V ,W ) where W is a (pseudo)-reflection group.
Try at least to build “unipotent characters” of G, or at least to buildtheir degrees (polynomials in x), Frobenius eigenvalues (roots ofunity), Fourier matrices.
I Lusztig knew already a solution for Coxeter groups which are not Weylgroups (except the Fourier matrix for H4 which was determined byMalle in 1994).
I Malle gave a solution for imprimitive spetsial complex reflectiongroups in 1995.
I Stating a long series of precise axioms — many of technical nature —we can now show that there is a unique solution for all primitive
spetsial complex reflection groups.
M. Broue, G. Malle, J. Michel, Split spetses for primitive reflection groups, Asterisque 359 (2014)
Michel Broue About Spetses
![Page 99: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/99.jpg)
II. Towards Spetses
Try to treat a complex reflection group as a Weyl group: try to builda thing G(x) (x an indeterminate) associated with a typeG = (V ,W ) where W is a (pseudo)-reflection group.
Try at least to build “unipotent characters” of G, or at least to buildtheir degrees (polynomials in x), Frobenius eigenvalues (roots ofunity), Fourier matrices.
I Lusztig knew already a solution for Coxeter groups which are not Weylgroups (except the Fourier matrix for H4 which was determined byMalle in 1994).
I Malle gave a solution for imprimitive spetsial complex reflectiongroups in 1995.
I Stating a long series of precise axioms — many of technical nature —we can now show that there is a unique solution for all primitive
spetsial complex reflection groups.
M. Broue, G. Malle, J. Michel, Split spetses for primitive reflection groups, Asterisque 359 (2014)
Michel Broue About Spetses
![Page 100: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/100.jpg)
II. Towards Spetses
Try to treat a complex reflection group as a Weyl group: try to builda thing G(x) (x an indeterminate) associated with a typeG = (V ,W ) where W is a (pseudo)-reflection group.
Try at least to build “unipotent characters” of G, or at least to buildtheir degrees (polynomials in x), Frobenius eigenvalues (roots ofunity), Fourier matrices.
I Lusztig knew already a solution for Coxeter groups which are not Weylgroups (except the Fourier matrix for H4 which was determined byMalle in 1994).
I Malle gave a solution for imprimitive spetsial complex reflectiongroups in 1995.
I Stating a long series of precise axioms — many of technical nature —we can now show that there is a unique solution for all primitive
spetsial complex reflection groups.
M. Broue, G. Malle, J. Michel, Split spetses for primitive reflection groups, Asterisque 359 (2014)
Michel Broue About Spetses
![Page 101: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/101.jpg)
II. Towards Spetses
Try to treat a complex reflection group as a Weyl group: try to builda thing G(x) (x an indeterminate) associated with a typeG = (V ,W ) where W is a (pseudo)-reflection group.
Try at least to build “unipotent characters” of G, or at least to buildtheir degrees (polynomials in x), Frobenius eigenvalues (roots ofunity), Fourier matrices.
I Lusztig knew already a solution for Coxeter groups which are not Weylgroups (except the Fourier matrix for H4 which was determined byMalle in 1994).
I Malle gave a solution for imprimitive spetsial complex reflectiongroups in 1995.
I Stating a long series of precise axioms — many of technical nature —we can now show that there is a unique solution for all primitive
spetsial complex reflection groups.
M. Broue, G. Malle, J. Michel, Split spetses for primitive reflection groups, Asterisque 359 (2014)
Michel Broue About Spetses
![Page 102: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/102.jpg)
II. Towards Spetses
Try to treat a complex reflection group as a Weyl group: try to builda thing G(x) (x an indeterminate) associated with a typeG = (V ,W ) where W is a (pseudo)-reflection group.
Try at least to build “unipotent characters” of G, or at least to buildtheir degrees (polynomials in x), Frobenius eigenvalues (roots ofunity), Fourier matrices.
I Lusztig knew already a solution for Coxeter groups which are not Weylgroups (except the Fourier matrix for H4 which was determined byMalle in 1994).
I Malle gave a solution for imprimitive spetsial complex reflectiongroups in 1995.
I Stating a long series of precise axioms — many of technical nature —we can now show that there is a unique solution for all primitive
spetsial complex reflection groups.
M. Broue, G. Malle, J. Michel, Split spetses for primitive reflection groups, Asterisque 359 (2014)
Michel Broue About Spetses
![Page 103: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/103.jpg)
A double object : double N
G = (V ,W ), where
V is a complex vector space of dimension r ,
W is finite (pseudo)-reflection subgroup of GL(V ),
A(W ) := the hyperplanes arrangement of W .
NhypW := number of reflecting hyperplanes,
N refW := number of reflections.
NhypW = N ref
W if W is generated by true reflections.
Michel Broue About Spetses
![Page 104: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/104.jpg)
A double object : double N
G = (V ,W ), where
V is a complex vector space of dimension r ,
W is finite (pseudo)-reflection subgroup of GL(V ),
A(W ) := the hyperplanes arrangement of W .
NhypW := number of reflecting hyperplanes,
N refW := number of reflections.
NhypW = N ref
W if W is generated by true reflections.
Michel Broue About Spetses
![Page 105: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/105.jpg)
A double object : double N
G = (V ,W ), where
V is a complex vector space of dimension r ,
W is finite (pseudo)-reflection subgroup of GL(V ),
A(W ) := the hyperplanes arrangement of W .
NhypW := number of reflecting hyperplanes,
N refW := number of reflections.
NhypW = N ref
W if W is generated by true reflections.
Michel Broue About Spetses
![Page 106: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/106.jpg)
A double object : double N
G = (V ,W ), where
V is a complex vector space of dimension r ,
W is finite (pseudo)-reflection subgroup of GL(V ),
A(W ) := the hyperplanes arrangement of W .
NhypW := number of reflecting hyperplanes,
N refW := number of reflections.
NhypW = N ref
W if W is generated by true reflections.
Michel Broue About Spetses
![Page 107: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/107.jpg)
A double object : double N
G = (V ,W ), where
V is a complex vector space of dimension r ,
W is finite (pseudo)-reflection subgroup of GL(V ),
A(W ) := the hyperplanes arrangement of W .
NhypW := number of reflecting hyperplanes,
N refW := number of reflections.
NhypW = N ref
W if W is generated by true reflections.
Michel Broue About Spetses
![Page 108: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/108.jpg)
A double object : double N
G = (V ,W ), where
V is a complex vector space of dimension r ,
W is finite (pseudo)-reflection subgroup of GL(V ),
A(W ) := the hyperplanes arrangement of W .
NhypW := number of reflecting hyperplanes,
N refW := number of reflections.
NhypW = N ref
W if W is generated by true reflections.
Michel Broue About Spetses
![Page 109: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/109.jpg)
A double object : double N
G = (V ,W ), where
V is a complex vector space of dimension r ,
W is finite (pseudo)-reflection subgroup of GL(V ),
A(W ) := the hyperplanes arrangement of W .
NhypW := number of reflecting hyperplanes,
N refW := number of reflections.
NhypW = N ref
W if W is generated by true reflections.
Michel Broue About Spetses
![Page 110: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/110.jpg)
A double object : double N
G = (V ,W ), where
V is a complex vector space of dimension r ,
W is finite (pseudo)-reflection subgroup of GL(V ),
A(W ) := the hyperplanes arrangement of W .
NhypW := number of reflecting hyperplanes,
N refW := number of reflections.
NhypW = N ref
W if W is generated by true reflections.
Michel Broue About Spetses
![Page 111: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/111.jpg)
Double polynomial order
|Gc|(x) := (−1)rxNhypW
11
|W |∑
w∈W1
detV (1− wx)∗
|Gnc|(x) := (−1)rxNrefW
11
|W |∑
w∈W1
detV (1− wx)∗
The compact and the noncompact order coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 112: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/112.jpg)
Double polynomial order
|Gc|(x) := (−1)rxNhypW
11
|W |∑
w∈W1
detV (1− wx)∗
|Gnc|(x) := (−1)rxNrefW
11
|W |∑
w∈W1
detV (1− wx)∗
The compact and the noncompact order coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 113: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/113.jpg)
Double polynomial order
|Gc|(x) := (−1)rxNhypW
11
|W |∑
w∈W1
detV (1− wx)∗
|Gnc|(x) := (−1)rxNrefW
11
|W |∑
w∈W1
detV (1− wx)∗
The compact and the noncompact order coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 114: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/114.jpg)
2. Spetsial ζ-cyclotomic Hecke algebras for G
As above,
Wζ = CW (w) is the centralizer of a ζ-regular element w ∈W ,
A ζ-cyclotomic Hecke algebra H(Wζ) is spetsial for G if
1 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1] ,
2 and H(Wζ) satisfies various technical conditions, which split intoF compact type conditions,F noncompact type conditions.
These conditions coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 115: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/115.jpg)
2. Spetsial ζ-cyclotomic Hecke algebras for G
As above,
Wζ = CW (w) is the centralizer of a ζ-regular element w ∈W ,
A ζ-cyclotomic Hecke algebra H(Wζ) is spetsial for G if
1 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1] ,
2 and H(Wζ) satisfies various technical conditions, which split intoF compact type conditions,F noncompact type conditions.
These conditions coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 116: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/116.jpg)
2. Spetsial ζ-cyclotomic Hecke algebras for G
As above,
Wζ = CW (w) is the centralizer of a ζ-regular element w ∈W ,
A ζ-cyclotomic Hecke algebra H(Wζ) is spetsial for G if
1 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1] ,
2 and H(Wζ) satisfies various technical conditions, which split intoF compact type conditions,F noncompact type conditions.
These conditions coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 117: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/117.jpg)
2. Spetsial ζ-cyclotomic Hecke algebras for G
As above,
Wζ = CW (w) is the centralizer of a ζ-regular element w ∈W ,
A ζ-cyclotomic Hecke algebra H(Wζ) is spetsial for G if
1 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1] ,
2 and H(Wζ) satisfies various technical conditions, which split intoF compact type conditions,F noncompact type conditions.
These conditions coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 118: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/118.jpg)
2. Spetsial ζ-cyclotomic Hecke algebras for G
As above,
Wζ = CW (w) is the centralizer of a ζ-regular element w ∈W ,
A ζ-cyclotomic Hecke algebra H(Wζ) is spetsial for G if
1 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1] ,
2 and H(Wζ) satisfies various technical conditions, which split intoF compact type conditions,F noncompact type conditions.
These conditions coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 119: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/119.jpg)
2. Spetsial ζ-cyclotomic Hecke algebras for G
As above,
Wζ = CW (w) is the centralizer of a ζ-regular element w ∈W ,
A ζ-cyclotomic Hecke algebra H(Wζ) is spetsial for G if
1 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1] ,
2 and H(Wζ) satisfies various technical conditions, which split into
F compact type conditions,F noncompact type conditions.
These conditions coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 120: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/120.jpg)
2. Spetsial ζ-cyclotomic Hecke algebras for G
As above,
Wζ = CW (w) is the centralizer of a ζ-regular element w ∈W ,
A ζ-cyclotomic Hecke algebra H(Wζ) is spetsial for G if
1 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1] ,
2 and H(Wζ) satisfies various technical conditions, which split intoF compact type conditions,
F noncompact type conditions.
These conditions coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 121: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/121.jpg)
2. Spetsial ζ-cyclotomic Hecke algebras for G
As above,
Wζ = CW (w) is the centralizer of a ζ-regular element w ∈W ,
A ζ-cyclotomic Hecke algebra H(Wζ) is spetsial for G if
1 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1] ,
2 and H(Wζ) satisfies various technical conditions, which split intoF compact type conditions,F noncompact type conditions.
These conditions coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 122: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/122.jpg)
2. Spetsial ζ-cyclotomic Hecke algebras for G
As above,
Wζ = CW (w) is the centralizer of a ζ-regular element w ∈W ,
A ζ-cyclotomic Hecke algebra H(Wζ) is spetsial for G if
1 for each absolute irreducible character χ of H(Wζ),
Sχ ∈ C[x , x−1] ,
2 and H(Wζ) satisfies various technical conditions, which split intoF compact type conditions,F noncompact type conditions.
These conditions coincide if W is generated by true reflections.
Michel Broue About Spetses
![Page 123: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/123.jpg)
2.1. Spetsial 1-cyclotomic Hecke algebras, special groups
Theorem
1 A 1-cyclotomic Hecke algebra can be spetsial of compact type for Gonly if it is the algebra Hc(W ) defined by
Hc(W ) = 〈sH〉H∈A with relations:
(sH − x)(1 + sH+ · · ·+ seH−1H ) = 0 (if sH has order eH)
2 A 1-cyclotomic Hecke algebras can be spetsial of noncompact typefor G only if it is the algebra Hnc(W ) defined by
Hnc(W ) = 〈sH〉H∈A with relations:
(sH − x)(xeH−1 + xeH−2sH + · · ·+ seH−1H ) = 0 .
Michel Broue About Spetses
![Page 124: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/124.jpg)
2.1. Spetsial 1-cyclotomic Hecke algebras, special groups
Theorem1 A 1-cyclotomic Hecke algebra can be spetsial of compact type for G
only if it is the algebra Hc(W ) defined byHc(W ) = 〈sH〉H∈A with relations:
(sH − x)(1 + sH+ · · ·+ seH−1H ) = 0 (if sH has order eH)
2 A 1-cyclotomic Hecke algebras can be spetsial of noncompact typefor G only if it is the algebra Hnc(W ) defined by
Hnc(W ) = 〈sH〉H∈A with relations:
(sH − x)(xeH−1 + xeH−2sH + · · ·+ seH−1H ) = 0 .
Michel Broue About Spetses
![Page 125: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/125.jpg)
2.1. Spetsial 1-cyclotomic Hecke algebras, special groups
Theorem1 A 1-cyclotomic Hecke algebra can be spetsial of compact type for G
only if it is the algebra Hc(W ) defined byHc(W ) = 〈sH〉H∈A with relations:
(sH − x)(1 + sH+ · · ·+ seH−1H ) = 0 (if sH has order eH)
2 A 1-cyclotomic Hecke algebras can be spetsial of noncompact typefor G only if it is the algebra Hnc(W ) defined by
Hnc(W ) = 〈sH〉H∈A with relations:
(sH − x)(xeH−1 + xeH−2sH + · · ·+ seH−1H ) = 0 .
Michel Broue About Spetses
![Page 126: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/126.jpg)
2.2. Spetsial groups
Let H(W ) denote either Hc(W ) or Hnc(W ).
Theorem (G. Malle)–Definition
Assume W acts irreducibly on V . The following assertions are equivalent.
(i) H(W ) is spetsial.
(ii) For each absolutely irreducible character χ of H(W ), Sχ ∈ C[x , x−1] .
(iii) W is one of the following groups (Shephard–Todd’s notation), calledthe spetsial groups :
G (d , 1, n)(d ,n≥1) , G (e, e, n)(e,n≥2),
all groups Gi (4 ≤ i ≤ 37) well generated by true reflections,
G4, G6, G8, G25, G26, G32 .
Michel Broue About Spetses
![Page 127: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/127.jpg)
2.2. Spetsial groups
Let H(W ) denote either Hc(W ) or Hnc(W ).
Theorem (G. Malle)–Definition
Assume W acts irreducibly on V . The following assertions are equivalent.
(i) H(W ) is spetsial.
(ii) For each absolutely irreducible character χ of H(W ), Sχ ∈ C[x , x−1] .
(iii) W is one of the following groups (Shephard–Todd’s notation), calledthe spetsial groups :
G (d , 1, n)(d ,n≥1) , G (e, e, n)(e,n≥2),
all groups Gi (4 ≤ i ≤ 37) well generated by true reflections,
G4, G6, G8, G25, G26, G32 .
Michel Broue About Spetses
![Page 128: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/128.jpg)
2.2. Spetsial groups
Let H(W ) denote either Hc(W ) or Hnc(W ).
Theorem (G. Malle)–Definition
Assume W acts irreducibly on V . The following assertions are equivalent.
(i) H(W ) is spetsial.
(ii) For each absolutely irreducible character χ of H(W ), Sχ ∈ C[x , x−1] .
(iii) W is one of the following groups (Shephard–Todd’s notation), calledthe spetsial groups :
G (d , 1, n)(d ,n≥1) , G (e, e, n)(e,n≥2),
all groups Gi (4 ≤ i ≤ 37) well generated by true reflections,
G4, G6, G8, G25, G26, G32 .
Michel Broue About Spetses
![Page 129: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/129.jpg)
2.2. Spetsial groups
Let H(W ) denote either Hc(W ) or Hnc(W ).
Theorem (G. Malle)–Definition
Assume W acts irreducibly on V . The following assertions are equivalent.
(i) H(W ) is spetsial.
(ii) For each absolutely irreducible character χ of H(W ), Sχ ∈ C[x , x−1] .
(iii) W is one of the following groups (Shephard–Todd’s notation), calledthe spetsial groups :
G (d , 1, n)(d ,n≥1) , G (e, e, n)(e,n≥2),
all groups Gi (4 ≤ i ≤ 37) well generated by true reflections,
G4, G6, G8, G25, G26, G32 .
Michel Broue About Spetses
![Page 130: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/130.jpg)
2.2. Spetsial groups
Let H(W ) denote either Hc(W ) or Hnc(W ).
Theorem (G. Malle)–Definition
Assume W acts irreducibly on V . The following assertions are equivalent.
(i) H(W ) is spetsial.
(ii) For each absolutely irreducible character χ of H(W ), Sχ ∈ C[x , x−1] .
(iii) W is one of the following groups (Shephard–Todd’s notation), calledthe spetsial groups :
G (d , 1, n)(d ,n≥1) , G (e, e, n)(e,n≥2),
all groups Gi (4 ≤ i ≤ 37) well generated by true reflections,
G4, G6, G8, G25, G26, G32 .
Michel Broue About Spetses
![Page 131: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/131.jpg)
2.2. Spetsial groups
Let H(W ) denote either Hc(W ) or Hnc(W ).
Theorem (G. Malle)–Definition
Assume W acts irreducibly on V . The following assertions are equivalent.
(i) H(W ) is spetsial.
(ii) For each absolutely irreducible character χ of H(W ), Sχ ∈ C[x , x−1] .
(iii) W is one of the following groups (Shephard–Todd’s notation), calledthe spetsial groups :
G (d , 1, n)(d ,n≥1) , G (e, e, n)(e,n≥2),
all groups Gi (4 ≤ i ≤ 37) well generated by true reflections,
G4, G6, G8, G25, G26, G32 .
Michel Broue About Spetses
![Page 132: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/132.jpg)
2.2. Spetsial groups
Let H(W ) denote either Hc(W ) or Hnc(W ).
Theorem (G. Malle)–Definition
Assume W acts irreducibly on V . The following assertions are equivalent.
(i) H(W ) is spetsial.
(ii) For each absolutely irreducible character χ of H(W ), Sχ ∈ C[x , x−1] .
(iii) W is one of the following groups (Shephard–Todd’s notation), calledthe spetsial groups :
G (d , 1, n)(d ,n≥1) , G (e, e, n)(e,n≥2),
all groups Gi (4 ≤ i ≤ 37) well generated by true reflections,
G4, G6, G8, G25, G26, G32 .
Michel Broue About Spetses
![Page 133: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/133.jpg)
2.2. Spetsial groups
Let H(W ) denote either Hc(W ) or Hnc(W ).
Theorem (G. Malle)–Definition
Assume W acts irreducibly on V . The following assertions are equivalent.
(i) H(W ) is spetsial.
(ii) For each absolutely irreducible character χ of H(W ), Sχ ∈ C[x , x−1] .
(iii) W is one of the following groups (Shephard–Todd’s notation), calledthe spetsial groups :
G (d , 1, n)(d ,n≥1) , G (e, e, n)(e,n≥2),
all groups Gi (4 ≤ i ≤ 37) well generated by true reflections,
G4, G6, G8, G25, G26, G32 .
Michel Broue About Spetses
![Page 134: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/134.jpg)
2.2. Spetsial groups
Let H(W ) denote either Hc(W ) or Hnc(W ).
Theorem (G. Malle)–Definition
Assume W acts irreducibly on V . The following assertions are equivalent.
(i) H(W ) is spetsial.
(ii) For each absolutely irreducible character χ of H(W ), Sχ ∈ C[x , x−1] .
(iii) W is one of the following groups (Shephard–Todd’s notation), calledthe spetsial groups :
G (d , 1, n)(d ,n≥1) , G (e, e, n)(e,n≥2),
all groups Gi (4 ≤ i ≤ 37) well generated by true reflections,
G4, G6, G8, G25, G26, G32 .
Michel Broue About Spetses
![Page 135: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/135.jpg)
2.2. Spetsial groups
Let H(W ) denote either Hc(W ) or Hnc(W ).
Theorem (G. Malle)–Definition
Assume W acts irreducibly on V . The following assertions are equivalent.
(i) H(W ) is spetsial.
(ii) For each absolutely irreducible character χ of H(W ), Sχ ∈ C[x , x−1] .
(iii) W is one of the following groups (Shephard–Todd’s notation), calledthe spetsial groups :
G (d , 1, n)(d ,n≥1) , G (e, e, n)(e,n≥2),
all groups Gi (4 ≤ i ≤ 37) well generated by true reflections,
G4, G6, G8, G25, G26, G32 .
Michel Broue About Spetses
![Page 136: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/136.jpg)
3. Some data associated with spetsial groups
Given G = (V ,W ) where W is special, there are
the set Un(Gc) of unipotent characters (compact type),
the set Un(Gnc) of unipotent characters (noncompact type),
which coincide if W is generated by true reflections
each of them (denoted Un(G) below), endowed with two maps
the map degreeDeg : Un(G)→ C[x ] , ρ 7→ Degρ(x) ,
defined up to sign,
the map Frobenius eigenvalue ρ 7→ Frρ, where Frρ is a root of unity,
a bijection (Alvis–Curtis duality)
Un(Gc)→ Un(Gnc) , ρ 7→ ρnc ,
with lots of properties (axioms) described below.
Michel Broue About Spetses
![Page 137: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/137.jpg)
3. Some data associated with spetsial groups
Given G = (V ,W ) where W is special, there are
the set Un(Gc) of unipotent characters (compact type),
the set Un(Gnc) of unipotent characters (noncompact type),
which coincide if W is generated by true reflections
each of them (denoted Un(G) below), endowed with two maps
the map degreeDeg : Un(G)→ C[x ] , ρ 7→ Degρ(x) ,
defined up to sign,
the map Frobenius eigenvalue ρ 7→ Frρ, where Frρ is a root of unity,
a bijection (Alvis–Curtis duality)
Un(Gc)→ Un(Gnc) , ρ 7→ ρnc ,
with lots of properties (axioms) described below.
Michel Broue About Spetses
![Page 138: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/138.jpg)
3. Some data associated with spetsial groups
Given G = (V ,W ) where W is special, there are
the set Un(Gc) of unipotent characters (compact type),
the set Un(Gnc) of unipotent characters (noncompact type),
which coincide if W is generated by true reflections
each of them (denoted Un(G) below), endowed with two maps
the map degreeDeg : Un(G)→ C[x ] , ρ 7→ Degρ(x) ,
defined up to sign,
the map Frobenius eigenvalue ρ 7→ Frρ, where Frρ is a root of unity,
a bijection (Alvis–Curtis duality)
Un(Gc)→ Un(Gnc) , ρ 7→ ρnc ,
with lots of properties (axioms) described below.
Michel Broue About Spetses
![Page 139: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/139.jpg)
3. Some data associated with spetsial groups
Given G = (V ,W ) where W is special, there are
the set Un(Gc) of unipotent characters (compact type),
the set Un(Gnc) of unipotent characters (noncompact type),
which coincide if W is generated by true reflections
each of them (denoted Un(G) below), endowed with two maps
the map degreeDeg : Un(G)→ C[x ] , ρ 7→ Degρ(x) ,
defined up to sign,
the map Frobenius eigenvalue ρ 7→ Frρ, where Frρ is a root of unity,
a bijection (Alvis–Curtis duality)
Un(Gc)→ Un(Gnc) , ρ 7→ ρnc ,
with lots of properties (axioms) described below.
Michel Broue About Spetses
![Page 140: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/140.jpg)
3. Some data associated with spetsial groups
Given G = (V ,W ) where W is special, there are
the set Un(Gc) of unipotent characters (compact type),
the set Un(Gnc) of unipotent characters (noncompact type),
which coincide if W is generated by true reflections
each of them (denoted Un(G) below), endowed with two maps
the map degreeDeg : Un(G)→ C[x ] , ρ 7→ Degρ(x) ,
defined up to sign,
the map Frobenius eigenvalue ρ 7→ Frρ, where Frρ is a root of unity,
a bijection (Alvis–Curtis duality)
Un(Gc)→ Un(Gnc) , ρ 7→ ρnc ,
with lots of properties (axioms) described below.
Michel Broue About Spetses
![Page 141: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/141.jpg)
3. Some data associated with spetsial groups
Given G = (V ,W ) where W is special, there are
the set Un(Gc) of unipotent characters (compact type),
the set Un(Gnc) of unipotent characters (noncompact type),
which coincide if W is generated by true reflections
each of them (denoted Un(G) below), endowed with two maps
the map degreeDeg : Un(G)→ C[x ] , ρ 7→ Degρ(x) ,
defined up to sign,
the map Frobenius eigenvalue ρ 7→ Frρ, where Frρ is a root of unity,
a bijection (Alvis–Curtis duality)
Un(Gc)→ Un(Gnc) , ρ 7→ ρnc ,
with lots of properties (axioms) described below.
Michel Broue About Spetses
![Page 142: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/142.jpg)
3. Some data associated with spetsial groups
Given G = (V ,W ) where W is special, there are
the set Un(Gc) of unipotent characters (compact type),
the set Un(Gnc) of unipotent characters (noncompact type),
which coincide if W is generated by true reflections
each of them (denoted Un(G) below), endowed with two maps
the map degreeDeg : Un(G)→ C[x ] , ρ 7→ Degρ(x) ,
defined up to sign,
the map Frobenius eigenvalue ρ 7→ Frρ, where Frρ is a root of unity,
a bijection (Alvis–Curtis duality)
Un(Gc)→ Un(Gnc) , ρ 7→ ρnc ,
with lots of properties (axioms) described below.
Michel Broue About Spetses
![Page 143: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/143.jpg)
3. Some data associated with spetsial groups
Given G = (V ,W ) where W is special, there are
the set Un(Gc) of unipotent characters (compact type),
the set Un(Gnc) of unipotent characters (noncompact type),
which coincide if W is generated by true reflections
each of them (denoted Un(G) below), endowed with two maps
the map degreeDeg : Un(G)→ C[x ] , ρ 7→ Degρ(x) ,
defined up to sign,
the map Frobenius eigenvalue ρ 7→ Frρ, where Frρ is a root of unity,
a bijection (Alvis–Curtis duality)
Un(Gc)→ Un(Gnc) , ρ 7→ ρnc ,
with lots of properties (axioms) described below.
Michel Broue About Spetses
![Page 144: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/144.jpg)
3. Some data associated with spetsial groups
Given G = (V ,W ) where W is special, there are
the set Un(Gc) of unipotent characters (compact type),
the set Un(Gnc) of unipotent characters (noncompact type),
which coincide if W is generated by true reflections
each of them (denoted Un(G) below), endowed with two maps
the map degreeDeg : Un(G)→ C[x ] , ρ 7→ Degρ(x) ,
defined up to sign,
the map Frobenius eigenvalue ρ 7→ Frρ, where Frρ is a root of unity,
a bijection (Alvis–Curtis duality)
Un(Gc)→ Un(Gnc) , ρ 7→ ρnc ,
with lots of properties (axioms) described below.
Michel Broue About Spetses
![Page 145: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/145.jpg)
3. Some data associated with spetsial groups
Given G = (V ,W ) where W is special, there are
the set Un(Gc) of unipotent characters (compact type),
the set Un(Gnc) of unipotent characters (noncompact type),
which coincide if W is generated by true reflections
each of them (denoted Un(G) below), endowed with two maps
the map degreeDeg : Un(G)→ C[x ] , ρ 7→ Degρ(x) ,
defined up to sign,
the map Frobenius eigenvalue ρ 7→ Frρ, where Frρ is a root of unity,
a bijection (Alvis–Curtis duality)
Un(Gc)→ Un(Gnc) , ρ 7→ ρnc ,
with lots of properties (axioms) described below.
Michel Broue About Spetses
![Page 146: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/146.jpg)
3.1. First axioms
Connection compact / noncompact
1 Degρnc(x) = xNrefW Degρ(1/x)∗ , n! up to sign!
2 FrρFrρnc = 1 .
From now on we only describe the compact type case.
Definition
Let ζ ∈ µ.The ζ-principal series is
Un(G, ζ) := ρ ∈ Un(G) | Degρ(ζ) 6= 0 .
Michel Broue About Spetses
![Page 147: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/147.jpg)
3.1. First axioms
Connection compact / noncompact
1 Degρnc(x) = xNrefW Degρ(1/x)∗ , n! up to sign!
2 FrρFrρnc = 1 .
From now on we only describe the compact type case.
Definition
Let ζ ∈ µ.The ζ-principal series is
Un(G, ζ) := ρ ∈ Un(G) | Degρ(ζ) 6= 0 .
Michel Broue About Spetses
![Page 148: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/148.jpg)
3.1. First axioms
Connection compact / noncompact
1 Degρnc(x) = xNrefW Degρ(1/x)∗ ,
n! up to sign!
2 FrρFrρnc = 1 .
From now on we only describe the compact type case.
Definition
Let ζ ∈ µ.The ζ-principal series is
Un(G, ζ) := ρ ∈ Un(G) | Degρ(ζ) 6= 0 .
Michel Broue About Spetses
![Page 149: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/149.jpg)
3.1. First axioms
Connection compact / noncompact
1 Degρnc(x) = xNrefW Degρ(1/x)∗ , n! up to sign!
2 FrρFrρnc = 1 .
From now on we only describe the compact type case.
Definition
Let ζ ∈ µ.The ζ-principal series is
Un(G, ζ) := ρ ∈ Un(G) | Degρ(ζ) 6= 0 .
Michel Broue About Spetses
![Page 150: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/150.jpg)
3.1. First axioms
Connection compact / noncompact
1 Degρnc(x) = xNrefW Degρ(1/x)∗ , n! up to sign!
2 FrρFrρnc = 1 .
From now on we only describe the compact type case.
Definition
Let ζ ∈ µ.The ζ-principal series is
Un(G, ζ) := ρ ∈ Un(G) | Degρ(ζ) 6= 0 .
Michel Broue About Spetses
![Page 151: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/151.jpg)
3.1. First axioms
Connection compact / noncompact
1 Degρnc(x) = xNrefW Degρ(1/x)∗ , n! up to sign!
2 FrρFrρnc = 1 .
From now on we only describe the compact type case.
Definition
Let ζ ∈ µ.The ζ-principal series is
Un(G, ζ) := ρ ∈ Un(G) | Degρ(ζ) 6= 0 .
Michel Broue About Spetses
![Page 152: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/152.jpg)
3.1. First axioms
Connection compact / noncompact
1 Degρnc(x) = xNrefW Degρ(1/x)∗ , n! up to sign!
2 FrρFrρnc = 1 .
From now on we only describe the compact type case.
Definition
Let ζ ∈ µ.
The ζ-principal series is
Un(G, ζ) := ρ ∈ Un(G) | Degρ(ζ) 6= 0 .
Michel Broue About Spetses
![Page 153: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/153.jpg)
3.1. First axioms
Connection compact / noncompact
1 Degρnc(x) = xNrefW Degρ(1/x)∗ , n! up to sign!
2 FrρFrρnc = 1 .
From now on we only describe the compact type case.
Definition
Let ζ ∈ µ.The ζ-principal series is
Un(G, ζ) := ρ ∈ Un(G) | Degρ(ζ) 6= 0 .
Michel Broue About Spetses
![Page 154: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/154.jpg)
ζ-Axioms (compact type)
3.2. ζ-Axioms
For w ∈W a ζ-regular element, there are
a spetsial ζ-cyclotomic Hecke algebra of compact type H(Wζ)associated with w ,
and a bijection
IrrH(Wζ)∼−→ Un(G, ζ) , χ 7→ ρχ
such that
1 Degρχ(x) = ± [|G|(x) : |Tw |(x)]x ′
Sχ(x),
2 Frρχ = explicit formula depending only on H(Wζ) and χ .
Michel Broue About Spetses
![Page 155: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/155.jpg)
ζ-Axioms (compact type)
3.2. ζ-Axioms
For w ∈W a ζ-regular element, there are
a spetsial ζ-cyclotomic Hecke algebra of compact type H(Wζ)associated with w ,
and a bijection
IrrH(Wζ)∼−→ Un(G, ζ) , χ 7→ ρχ
such that
1 Degρχ(x) = ± [|G|(x) : |Tw |(x)]x ′
Sχ(x),
2 Frρχ = explicit formula depending only on H(Wζ) and χ .
Michel Broue About Spetses
![Page 156: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/156.jpg)
ζ-Axioms (compact type)
3.2. ζ-Axioms
For w ∈W a ζ-regular element, there are
a spetsial ζ-cyclotomic Hecke algebra of compact type H(Wζ)associated with w ,
and a bijection
IrrH(Wζ)∼−→ Un(G, ζ) , χ 7→ ρχ
such that
1 Degρχ(x) = ± [|G|(x) : |Tw |(x)]x ′
Sχ(x),
2 Frρχ = explicit formula depending only on H(Wζ) and χ .
Michel Broue About Spetses
![Page 157: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/157.jpg)
ζ-Axioms (compact type)
3.2. ζ-Axioms
For w ∈W a ζ-regular element, there are
a spetsial ζ-cyclotomic Hecke algebra of compact type H(Wζ)associated with w ,
and a bijection
IrrH(Wζ)∼−→ Un(G, ζ) , χ 7→ ρχ
such that
1 Degρχ(x) = ± [|G|(x) : |Tw |(x)]x ′
Sχ(x),
2 Frρχ = explicit formula depending only on H(Wζ) and χ .
Michel Broue About Spetses
![Page 158: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/158.jpg)
ζ-Axioms (compact type)
3.2. ζ-Axioms
For w ∈W a ζ-regular element, there are
a spetsial ζ-cyclotomic Hecke algebra of compact type H(Wζ)associated with w ,
and a bijection
IrrH(Wζ)∼−→ Un(G, ζ) , χ 7→ ρχ
such that
1 Degρχ(x) = ± [|G|(x) : |Tw |(x)]x ′
Sχ(x),
2 Frρχ = explicit formula depending only on H(Wζ) and χ .
Michel Broue About Spetses
![Page 159: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/159.jpg)
ζ-Axioms (compact type)
3.2. ζ-Axioms
For w ∈W a ζ-regular element, there are
a spetsial ζ-cyclotomic Hecke algebra of compact type H(Wζ)associated with w ,
and a bijection
IrrH(Wζ)∼−→ Un(G, ζ) , χ 7→ ρχ
such that
1 Degρχ(x) = ± [|G|(x) : |Tw |(x)]x ′
Sχ(x),
2 Frρχ = explicit formula depending only on H(Wζ) and χ .
Michel Broue About Spetses
![Page 160: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/160.jpg)
ζ-Axioms (compact type)
3.2. ζ-Axioms
For w ∈W a ζ-regular element, there are
a spetsial ζ-cyclotomic Hecke algebra of compact type H(Wζ)associated with w ,
and a bijection
IrrH(Wζ)∼−→ Un(G, ζ) , χ 7→ ρχ
such that
1 Degρχ(x) = ± [|G|(x) : |Tw |(x)]x ′
Sχ(x),
2 Frρχ = explicit formula depending only on H(Wζ) and χ .
Michel Broue About Spetses
![Page 161: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/161.jpg)
3.3. Rouquier blocks
If the representation of Wζ on Vζ is rational over some cyclotomicfield K , the ζ-cyclotomic Hecke algebra H(Wζ) may be defined overZK [x , x−1].
Definition
The Rouquier blocks of a ζ-cyclotomic Hecke algebra H(Wζ) are theblocks of the algebra
ZK [x , x−1,((xn − 1)−1
)n≥1
]⊗Z[x ,x−1] H(Wζ) .
The Rouquier blocks of ζ-cyclotomic Hecke algebras have beenclassified in all cases (Malle–Rouquier, B.–Kim, Chlouveraki).
For ζ = 1 and W Coxeter group, Rouquier blocks are nothing but thecharacters associated with two sided cells (Kazhdan–Lusztig theory).
Michel Broue About Spetses
![Page 162: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/162.jpg)
3.3. Rouquier blocks
If the representation of Wζ on Vζ is rational over some cyclotomicfield K , the ζ-cyclotomic Hecke algebra H(Wζ) may be defined overZK [x , x−1].
Definition
The Rouquier blocks of a ζ-cyclotomic Hecke algebra H(Wζ) are theblocks of the algebra
ZK [x , x−1,((xn − 1)−1
)n≥1
]⊗Z[x ,x−1] H(Wζ) .
The Rouquier blocks of ζ-cyclotomic Hecke algebras have beenclassified in all cases (Malle–Rouquier, B.–Kim, Chlouveraki).
For ζ = 1 and W Coxeter group, Rouquier blocks are nothing but thecharacters associated with two sided cells (Kazhdan–Lusztig theory).
Michel Broue About Spetses
![Page 163: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/163.jpg)
3.4. Families and Rouquier blocks
Families
There is a partition
Un(G) =⊔
F∈Fam(G)
F
(where the F ’s are the families of unipotent characters), hence for allregular ζ,
Un(G, ζ) =⊔
F∈Fam(G)
(F ∩ Un(G, ζ)) ,
with the following properties.
1 Through the bijection Un(G, ζ)∼−→ IrrH(Wζ) , the nonempty
intersections F ∩ Un(G, ζ) are the Rouquier blocks of IrrH(Wζ).
2 The integers aρ (valuation of Degρ) and Aρ (degree of Degρ) areconstant for ρ in a family F .
Michel Broue About Spetses
![Page 164: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/164.jpg)
3.4. Families and Rouquier blocks
Families
There is a partition
Un(G) =⊔
F∈Fam(G)
F
(where the F ’s are the families of unipotent characters), hence for allregular ζ,
Un(G, ζ) =⊔
F∈Fam(G)
(F ∩ Un(G, ζ)) ,
with the following properties.
1 Through the bijection Un(G, ζ)∼−→ IrrH(Wζ) , the nonempty
intersections F ∩ Un(G, ζ) are the Rouquier blocks of IrrH(Wζ).
2 The integers aρ (valuation of Degρ) and Aρ (degree of Degρ) areconstant for ρ in a family F .
Michel Broue About Spetses
![Page 165: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/165.jpg)
The Fourier matrices
Let us denote by B2 the braid group on three brands, generated by twoelements s and t satisfying the relation
s• t• sts = tst .
Let us set w0 := sts. It is known that
the center of B2 is infinite cyclic and generated byw2
0 = (sts)2 = (st)3,
the map
s 7→(
1 01 1
), t 7→
(1 −10 1
)induces an isomorphism B2/〈w4
0〉∼−→ SL2(Z).
Michel Broue About Spetses
![Page 166: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/166.jpg)
The Fourier matrices
Let us denote by B2 the braid group on three brands, generated by twoelements s and t satisfying the relation
s• t• sts = tst .
Let us set w0 := sts. It is known that
the center of B2 is infinite cyclic and generated byw2
0 = (sts)2 = (st)3,
the map
s 7→(
1 01 1
), t 7→
(1 −10 1
)induces an isomorphism B2/〈w4
0〉∼−→ SL2(Z).
Michel Broue About Spetses
![Page 167: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/167.jpg)
The Fourier matrices
Let us denote by B2 the braid group on three brands, generated by twoelements s and t satisfying the relation
s• t• sts = tst .
Let us set w0 := sts. It is known that
the center of B2 is infinite cyclic and generated byw2
0 = (sts)2 = (st)3,
the map
s 7→(
1 01 1
), t 7→
(1 −10 1
)induces an isomorphism B2/〈w4
0〉∼−→ SL2(Z).
Michel Broue About Spetses
![Page 168: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/168.jpg)
Let F be a family in Un(G).
The S-matric (Fourier matrix)
There is a complex matrix S with entries indexed by F × F , such that forall χ0 ∈ Irr(W ), ∑
χ∈Irr(W )
Sρχ,ρχ0Fegχ = Degρχ0
,
and with the following properties.
1 S is unitary and symmetric,
2 S2 is an order 2 monomial matrix with entries in ±1,3 there exists a special character of W (in the Rouquier block
corresponding to F) such that1 the corresponding row i0 of S has no zero entry,2 (Verlinde type formula) for all i , j , k ∈ F , the sums
∑l Sl,iSl,jS
∗l,kS
−1l,i0
are integers.
Michel Broue About Spetses
![Page 169: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/169.jpg)
Let F be a family in Un(G).
The S-matric (Fourier matrix)
There is a complex matrix S with entries indexed by F × F , such that forall χ0 ∈ Irr(W ), ∑
χ∈Irr(W )
Sρχ,ρχ0Fegχ = Degρχ0
,
and with the following properties.
1 S is unitary and symmetric,
2 S2 is an order 2 monomial matrix with entries in ±1,3 there exists a special character of W (in the Rouquier block
corresponding to F) such that1 the corresponding row i0 of S has no zero entry,2 (Verlinde type formula) for all i , j , k ∈ F , the sums
∑l Sl,iSl,jS
∗l,kS
−1l,i0
are integers.
Michel Broue About Spetses
![Page 170: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/170.jpg)
Let F be a family in Un(G).
The S-matric (Fourier matrix)
There is a complex matrix S with entries indexed by F × F , such that forall χ0 ∈ Irr(W ), ∑
χ∈Irr(W )
Sρχ,ρχ0Fegχ = Degρχ0
,
and with the following properties.
1 S is unitary and symmetric,
2 S2 is an order 2 monomial matrix with entries in ±1,3 there exists a special character of W (in the Rouquier block
corresponding to F) such that1 the corresponding row i0 of S has no zero entry,2 (Verlinde type formula) for all i , j , k ∈ F , the sums
∑l Sl,iSl,jS
∗l,kS
−1l,i0
are integers.
Michel Broue About Spetses
![Page 171: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/171.jpg)
Let F be a family in Un(G).
The S-matric (Fourier matrix)
There is a complex matrix S with entries indexed by F × F , such that forall χ0 ∈ Irr(W ), ∑
χ∈Irr(W )
Sρχ,ρχ0Fegχ = Degρχ0
,
and with the following properties.
1 S is unitary and symmetric,
2 S2 is an order 2 monomial matrix with entries in ±1,3 there exists a special character of W (in the Rouquier block
corresponding to F) such that1 the corresponding row i0 of S has no zero entry,2 (Verlinde type formula) for all i , j , k ∈ F , the sums
∑l Sl,iSl,jS
∗l,kS
−1l,i0
are integers.
Michel Broue About Spetses
![Page 172: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/172.jpg)
Let F be a family in Un(G).
The S-matric (Fourier matrix)
There is a complex matrix S with entries indexed by F × F , such that forall χ0 ∈ Irr(W ), ∑
χ∈Irr(W )
Sρχ,ρχ0Fegχ = Degρχ0
,
and with the following properties.
1 S is unitary and symmetric,
2 S2 is an order 2 monomial matrix with entries in ±1,3 there exists a special character of W (in the Rouquier block
corresponding to F) such that1 the corresponding row i0 of S has no zero entry,2 (Verlinde type formula) for all i , j , k ∈ F , the sums
∑l Sl,iSl,jS
∗l,kS
−1l,i0
are integers.
Michel Broue About Spetses
![Page 173: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/173.jpg)
Let F be a family in Un(G).
The S-matric (Fourier matrix)
There is a complex matrix S with entries indexed by F × F , such that forall χ0 ∈ Irr(W ), ∑
χ∈Irr(W )
Sρχ,ρχ0Fegχ = Degρχ0
,
and with the following properties.
1 S is unitary and symmetric,
2 S2 is an order 2 monomial matrix with entries in ±1,
3 there exists a special character of W (in the Rouquier blockcorresponding to F) such that
1 the corresponding row i0 of S has no zero entry,2 (Verlinde type formula) for all i , j , k ∈ F , the sums
∑l Sl,iSl,jS
∗l,kS
−1l,i0
are integers.
Michel Broue About Spetses
![Page 174: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/174.jpg)
Let F be a family in Un(G).
The S-matric (Fourier matrix)
There is a complex matrix S with entries indexed by F × F , such that forall χ0 ∈ Irr(W ), ∑
χ∈Irr(W )
Sρχ,ρχ0Fegχ = Degρχ0
,
and with the following properties.
1 S is unitary and symmetric,
2 S2 is an order 2 monomial matrix with entries in ±1,3 there exists a special character of W (in the Rouquier block
corresponding to F) such that
1 the corresponding row i0 of S has no zero entry,2 (Verlinde type formula) for all i , j , k ∈ F , the sums
∑l Sl,iSl,jS
∗l,kS
−1l,i0
are integers.
Michel Broue About Spetses
![Page 175: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/175.jpg)
Let F be a family in Un(G).
The S-matric (Fourier matrix)
There is a complex matrix S with entries indexed by F × F , such that forall χ0 ∈ Irr(W ), ∑
χ∈Irr(W )
Sρχ,ρχ0Fegχ = Degρχ0
,
and with the following properties.
1 S is unitary and symmetric,
2 S2 is an order 2 monomial matrix with entries in ±1,3 there exists a special character of W (in the Rouquier block
corresponding to F) such that1 the corresponding row i0 of S has no zero entry,
2 (Verlinde type formula) for all i , j , k ∈ F , the sums∑
l Sl,iSl,jS∗l,kS
−1l,i0
are integers.
Michel Broue About Spetses
![Page 176: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/176.jpg)
Let F be a family in Un(G).
The S-matric (Fourier matrix)
There is a complex matrix S with entries indexed by F × F , such that forall χ0 ∈ Irr(W ), ∑
χ∈Irr(W )
Sρχ,ρχ0Fegχ = Degρχ0
,
and with the following properties.
1 S is unitary and symmetric,
2 S2 is an order 2 monomial matrix with entries in ±1,3 there exists a special character of W (in the Rouquier block
corresponding to F) such that1 the corresponding row i0 of S has no zero entry,2 (Verlinde type formula) for all i , j , k ∈ F , the sums
∑l Sl,iSl,jS
∗l,kS
−1l,i0
are integers.
Michel Broue About Spetses
![Page 177: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/177.jpg)
Frobenius and Shintani matrices
Let Ω be the diagonal matrix indexed by F × F whose diagonal termat ρ ∈ F is the Frobenius eigenvalue Frρ.
Define Sh := S · Ω · S−1 .
Facta
a“There is a proof, but so far I’ve not seen an explanation” [JHC]
The maps 7→ Ω , t 7→ Sh
induces a representation of SL2(Z) onto the complex vector space withbasis F such that w0 7→ S .
All this makes us think of a kind of modular datum, and perhaps for theSpets of a kind of triangulated modular tensor category (?).
Michel Broue About Spetses
![Page 178: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/178.jpg)
Frobenius and Shintani matrices
Let Ω be the diagonal matrix indexed by F × F whose diagonal termat ρ ∈ F is the Frobenius eigenvalue Frρ.
Define Sh := S · Ω · S−1 .
Facta
a“There is a proof, but so far I’ve not seen an explanation” [JHC]
The maps 7→ Ω , t 7→ Sh
induces a representation of SL2(Z) onto the complex vector space withbasis F such that w0 7→ S .
All this makes us think of a kind of modular datum, and perhaps for theSpets of a kind of triangulated modular tensor category (?).
Michel Broue About Spetses
![Page 179: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/179.jpg)
Frobenius and Shintani matrices
Let Ω be the diagonal matrix indexed by F × F whose diagonal termat ρ ∈ F is the Frobenius eigenvalue Frρ.
Define Sh := S · Ω · S−1 .
Facta
a“There is a proof, but so far I’ve not seen an explanation” [JHC]
The maps 7→ Ω , t 7→ Sh
induces a representation of SL2(Z) onto the complex vector space withbasis F such that w0 7→ S .
All this makes us think of a kind of modular datum, and perhaps for theSpets of a kind of triangulated modular tensor category (?).
Michel Broue About Spetses
![Page 180: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/180.jpg)
Frobenius and Shintani matrices
Let Ω be the diagonal matrix indexed by F × F whose diagonal termat ρ ∈ F is the Frobenius eigenvalue Frρ.
Define Sh := S · Ω · S−1 .
Facta
a“There is a proof, but so far I’ve not seen an explanation” [JHC]
The maps 7→ Ω , t 7→ Sh
induces a representation of SL2(Z) onto the complex vector space withbasis F such that w0 7→ S .
All this makes us think of a kind of modular datum, and perhaps for theSpets of a kind of triangulated modular tensor category (?).
Michel Broue About Spetses
![Page 181: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/181.jpg)
Frobenius and Shintani matrices
Let Ω be the diagonal matrix indexed by F × F whose diagonal termat ρ ∈ F is the Frobenius eigenvalue Frρ.
Define Sh := S · Ω · S−1 .
Facta
a“There is a proof, but so far I’ve not seen an explanation” [JHC]
The maps 7→ Ω , t 7→ Sh
induces a representation of SL2(Z) onto the complex vector space withbasis F such that w0 7→ S .
All this makes us think of a kind of modular datum, and perhaps for theSpets of a kind of triangulated modular tensor category (?).
Michel Broue About Spetses
![Page 182: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/182.jpg)
Frobenius and Shintani matrices
Let Ω be the diagonal matrix indexed by F × F whose diagonal termat ρ ∈ F is the Frobenius eigenvalue Frρ.
Define Sh := S · Ω · S−1 .
Facta
a“There is a proof, but so far I’ve not seen an explanation” [JHC]
The maps 7→ Ω , t 7→ Sh
induces a representation of SL2(Z) onto the complex vector space withbasis F such that w0 7→ S .
All this makes us think of a kind of modular datum, and perhaps for theSpets of a kind of triangulated modular tensor category (?).
Michel Broue About Spetses
![Page 183: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/183.jpg)
The Fourier matrix for G4
01 02 12 01 34 04 25 13
1 . . . . . . . . .
01 .1+ 1√
−3
2
1− 1√−3
2−1√−3
. . . . . .
02 .1− 1√
−3
2
1+ 1√−3
21√−3
. . . . . .
12 . −1√−3
1√−3
−1√−3
. . . . . .
. . . . 1 . . . . .
01 . . . . . 12√−3
−12√−3
12
−1√−3
12
34 . . . . . −12√−3
12√−3
12
1√−3
12
04 . . . . . 12
12
12 . −1
2
25 . . . . . −1√−3
1√−3
. −1√−3
.
13 . . . . . 12
12 −1
2 . 12
Michel Broue About Spetses
![Page 184: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/184.jpg)
Unipotent characters for G4 ©3 ©3
In red = the Φ′6–series.• = the Φ4–series.
Character Degree FakeDegree Eigenvalue Family
• φ1,0 • 1 1 1 C1
φ2,13−√−3
6 qΦ′3Φ4Φ′′6 qΦ4 1 X3.01
φ2,33+√−3
6 qΦ′′3Φ4Φ′6 q3Φ4 1 X3.02
Z3 : 2√−33 qΦ1Φ2Φ4 0 ζ2
3 X3.12• φ3,2 • q2Φ3Φ6 q2Φ3Φ6 1 C1
φ1,4−√−3
6 q4Φ′′3Φ4Φ′′6 q4 1 X5.1
φ1,8
√−36 q4Φ′3Φ4Φ′6 q8 1 X5.2
• φ2,5 • 12q
4Φ22Φ6 q5Φ4 1 X5.3
Z3 : 11√−33 q4Φ1Φ2Φ4 0 ζ2
3 X5.4• G4 • 1
2q4Φ2
1Φ3 0 −1 X5.5
Φ′3,Φ′′3 (resp. Φ′6,Φ
′′6 ) are factors of Φ3 (resp Φ6) in Q(ζ3)
Michel Broue About Spetses
![Page 185: About Spetses - Departamento de Matematicacms.dm.uba.ar/actividades/congresos/XXICLA/SlidesBroue.pdf · 2016. 7. 25. · About Spetses Michel Brou e Universit e Paris-Diderot Paris](https://reader033.vdocuments.us/reader033/viewer/2022061001/60b08a1abaa8735c0a4917d6/html5/thumbnails/185.jpg)
Unipotent characters for G4 ©3 ©3In red = the Φ′6–series.• = the Φ4–series.
Character Degree FakeDegree Eigenvalue Family
• φ1,0 • 1 1 1 C1
φ2,13−√−3
6 qΦ′3Φ4Φ′′6 qΦ4 1 X3.01
φ2,33+√−3
6 qΦ′′3Φ4Φ′6 q3Φ4 1 X3.02
Z3 : 2√−33 qΦ1Φ2Φ4 0 ζ2
3 X3.12• φ3,2 • q2Φ3Φ6 q2Φ3Φ6 1 C1
φ1,4−√−3
6 q4Φ′′3Φ4Φ′′6 q4 1 X5.1
φ1,8
√−36 q4Φ′3Φ4Φ′6 q8 1 X5.2
• φ2,5 • 12q
4Φ22Φ6 q5Φ4 1 X5.3
Z3 : 11√−33 q4Φ1Φ2Φ4 0 ζ2
3 X5.4• G4 • 1
2q4Φ2
1Φ3 0 −1 X5.5
Φ′3,Φ′′3 (resp. Φ′6,Φ
′′6 ) are factors of Φ3 (resp Φ6) in Q(ζ3)
Michel Broue About Spetses